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Chapter 27 Solutions
EBK NUMERICAL METHODS FOR ENGINEERS
- b. Solve the following higher order ODE using Laplace Transformation for y(t). y" + 3y" + 7y + 5y = 0, with y(0) = 1, y'(0) = 0, %3Darrow_forward(75 A B/s . ? A This option Question * If we use the method of variation of parameters, the particular solution of the following differential equation y" = 5x is: Yp(x) = ?x³ O This option Yp (x) = 5 %3| -x4 24arrow_forwardThe mass and stiffness matrix of the system is given by: M = and m2 k, +k, -k, K = -k, k, +k, Take m1 =10 kg, m2-5 kg, k1=k3=100 N/m, k2=84 N/m Use modal analysis and determi ne: a) The normalized stiffness ki b) Its eigenvalues and eigenvectorsarrow_forward
- 12. Start with Equation IVb.2.8 and obtain Equation IVb.2.9. For this purpose, first ignore the non-linear term compared with the two dominant terms. Then sub- stitute for the velocity and temperature profiles. To develop the integral, consider a case where the hydrodynamic boundary layer is thicker than the thermal bound- ary layer (thus the integral is zero for y > 8'). Arrange the result in terms of = 8'/8 and ignore *.arrow_forwardThe velocity, v, of a falling parachutist is given by v= (1-em), 8. -(c/m)t C where g = 9.8067 m/s². Given the mass, m of the parachutist is 70kg, velocity, v = 40 m/s at time t = 10 s, find the drag coefficient, c by using the Bisection method.arrow_forwardQ.No.5: (a) Develop an equation by using pi-theorem for the shear stress in a fluid flowing in a pipe assuming that the stress is a function of the diameter and the roughness of the pipe, and the density, viscosity and velocity of the fluid. (b) A model of a torpedo is tested in a towing tank at a velocity of 24.4 m/s. The prototype is expected to attain a velocity of 246 m/min in 15.6°C water. (i) What model scale has been used? (ii) What would be the model speed it tested in a wind tunnel under a pressure of 290 psi and at constant temperature 26.77°C?arrow_forward
- Derive the rule-of-mixtures expression for the composite extensional modulus E₁ assuming the existence of an interphase region. The starting point for the derivation would be the model shown below. For simplicity, assume the interphase, like the matrix, is isotropic with modulus E¹. With an interphase region there is a volume fraction associated with the interphase (i.e.,V;). For this situation: vf + vm + Vi = 1 wi |||||||arrow_forward(3) For the given boundary value problem, the exact solution is given as = 3x - 7y. (a) Based on the exact solution, find the values on all sides, (b) discretize the domain into 16 elements and 15 evenly spaced nodes. Run poisson.m and check if the finite element approximation and exact solution matches, (c) plot the D values from step (b) using topo.m. y Side 3 Side 1 8.0 (4) The temperature distribution in a flat slab needs to be studied under the conditions shown i the table. The ? in table indicates insulated boundary and Q is the distributed heat source. I all cases assume the upper and lower boundaries are insulated. Assume that the units of length energy, and temperature for the values shown are consistent with a unit value for the coefficier of thermal conductivity. Boundary Temperatures 6 Case A C D. D. 00 LEGION Side 4 z episarrow_forwardAn object is shot upward from the ground with an initial velocity of 640 ft/sec, and experiencés a constant deceleration of 32 ft/sec² due to gravity as well as a deceleration of (v(t) / 10) ft/sec due to air resistance, where v(t) is the object's velocity in ft/sec. (a) Set up and solve an initial-value problem to determine the object's velocity v(t) at time t. (b) At what time does the object reach its highest point?arrow_forward
- A two dimensional square plate (with 2m on each side) is subjected to the boundary conditions shown below. y T= 300 °C T= 70 °C 3 m T= 70°C 3 m T= 70 °C 1) Plot the temperature distribution obtained by the numerical solution a. using a uniform grid size of 0.5 m (Ax=Ay=0.5) b. using a uniform grid size of 0.1 m (Ax=Ay=0.1) 2) Plot temperatures (obtained by exact and two numerical solutions) as a function of a. x at y=1.0 m b. x at y=1.5 m c. y at x-1.0 m d. y at x-1.5 marrow_forwardA// Use Implicit Method to solve the temperature distribution of a long thin rod with a length of 9 cm and following values: k = 0.49 cal/(s cm °C), Ax = 3 cm, and At = 0.2 s. At t=0 s, the temperature of the rod is 10°C and the boundary conditions are fixed dT (9,t) 1 °C/cm. Note that the rod for alltimes at 7(0,t) = 80°C and derivative condition dx is aluminum with C = 0.2174 cal/g °C) and p = 2.7 g/cm³. Find the temperature values on the inner grid points and the right boundary for t = 0.4 s.arrow_forwardQ2/A/ Use the Crank-Nicolson method to solve for the temperature distribution of a long thin rod C with a length of 10 cm and the following values: k = 0.49 cal/(s cm °C), Ax = 2 cm, and At = st 0.1 s. Initially the temperature of the rod is 0°C and the boundary conditions are fixed for all times C=0.2174 cal/g °C) at 7(0, t) = 100°C and T(10, t) = 50°C. Note that the rod is aluminum with and = 2.7 g/cm³. List the tridiagonal system of equations and determined the temperature up P to 0.1 s.arrow_forward
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